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Mirrors > Home > MPE Home > Th. List > canthwdom | Structured version Visualization version GIF version |
Description: Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 9080, equivalent to canth 7314). (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
canthwdom | ⊢ ¬ 𝒫 𝐴 ≼* 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elpw 5315 | . . . . 5 ⊢ ∅ ∈ 𝒫 𝐴 | |
2 | ne0i 4298 | . . . . 5 ⊢ (∅ ∈ 𝒫 𝐴 → 𝒫 𝐴 ≠ ∅) | |
3 | 1, 2 | mp1i 13 | . . . 4 ⊢ (𝒫 𝐴 ≼* 𝐴 → 𝒫 𝐴 ≠ ∅) |
4 | brwdomn0 9513 | . . . 4 ⊢ (𝒫 𝐴 ≠ ∅ → (𝒫 𝐴 ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴)) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝒫 𝐴 ≼* 𝐴 → (𝒫 𝐴 ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴)) |
6 | 5 | ibi 267 | . 2 ⊢ (𝒫 𝐴 ≼* 𝐴 → ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴) |
7 | relwdom 9510 | . . . . 5 ⊢ Rel ≼* | |
8 | 7 | brrelex2i 5693 | . . . 4 ⊢ (𝒫 𝐴 ≼* 𝐴 → 𝐴 ∈ V) |
9 | foeq2 6757 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝑥)) | |
10 | pweq 4578 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
11 | foeq3 6758 | . . . . . . . 8 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → (𝑓:𝐴–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴)) | |
12 | 10, 11 | syl 17 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑓:𝐴–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴)) |
13 | 9, 12 | bitrd 279 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–onto→𝒫 𝑥 ↔ 𝑓:𝐴–onto→𝒫 𝐴)) |
14 | 13 | notbid 318 | . . . . 5 ⊢ (𝑥 = 𝐴 → (¬ 𝑓:𝑥–onto→𝒫 𝑥 ↔ ¬ 𝑓:𝐴–onto→𝒫 𝐴)) |
15 | vex 3451 | . . . . . 6 ⊢ 𝑥 ∈ V | |
16 | 15 | canth 7314 | . . . . 5 ⊢ ¬ 𝑓:𝑥–onto→𝒫 𝑥 |
17 | 14, 16 | vtoclg 3527 | . . . 4 ⊢ (𝐴 ∈ V → ¬ 𝑓:𝐴–onto→𝒫 𝐴) |
18 | 8, 17 | syl 17 | . . 3 ⊢ (𝒫 𝐴 ≼* 𝐴 → ¬ 𝑓:𝐴–onto→𝒫 𝐴) |
19 | 18 | nexdv 1940 | . 2 ⊢ (𝒫 𝐴 ≼* 𝐴 → ¬ ∃𝑓 𝑓:𝐴–onto→𝒫 𝐴) |
20 | 6, 19 | pm2.65i 193 | 1 ⊢ ¬ 𝒫 𝐴 ≼* 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2940 Vcvv 3447 ∅c0 4286 𝒫 cpw 4564 class class class wbr 5109 –onto→wfo 6498 ≼* cwdom 9508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fo 6506 df-fv 6508 df-wdom 9509 |
This theorem is referenced by: pwdjudom 10160 |
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